Your friend has 100 red marbles, 100 blue marbles and 2 jars.
He proposes a game. He fills the jars with the marbles, put the two jars behind his back and tell you to pick one of them at random. You’ll then close your eyes, he’ll hand you the jar you picked, and you’ll pick a random marble from that jar.
You win if the marble you pick is blue, and you lose otherwise.
To give you the best shot at winning, your friend gives you the two jars before the game starts and says you can move the marbles around however you’d like, as long as all 200 marbles are in one of the 2 jars (that is, you can’t throw any marbles away).
How should you move the marbles around to give yourself the best chance of picking a blue marble?
Put one blue marble in one jar, and put the rest of the marbles in the other jar. This will give you just about a 75% chance of picking a blue marble.
Let P(blue) = probability of picking a blue marble. Jar 1 contains 1 blue marble and Jar 2 contains 99 blue marbles and 100 red marbles.
So: P(blue) = P(jar 1) * P(blue in jar 1) + P(jar 2) * P(blue in jar 2)
P(blue) = 0.5 * 1 + 0.5 * 99/199
P(blue) = 0.748
Thus, we end up with ~75% chance of picking a blue marble.